Since the homotopy type of PU(ℋ)P U(\mathcal{H}) is that of an Eilenberg-MacLane spaceK(ℤ,2)K(\mathbb{Z},2), there is precisely one isomorphism class of such bundles representing a class α∈H3(X,ℤ)\alpha \in H^3(X, \mathbb{Z}).

Definition

The twisted K-theory with twist α∈H3(X,ℤ)\alpha \in H^3(X, \mathbb{Z}) is the set of homotopy-classes of sections of such a bundle

From the general nonsense of twisted cohomology this induces canonically now for every B2U(1)\mathbf{B}^2 U(1)-cocyclecc (for instance given by a bundle gerbe) a notion of cc-twisted VectrVectr-cohomology:

The seminal result on the relation to loop grouprepresentations, now again with twists in H0(X,ℤ2)×H1(X,ℤ2)×H3(X,ℤ)H^0(X,\mathbb{Z}_2) \times H^1(X,\mathbb{Z}_2) \times H^3(X, \mathbb{Z}), is in the series of articles